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I need to evaluate the following expression to determine the failure rate of an ensemble of classifiers. Basically it is the total probability of failure of more than 12 classifiers with each having error rate as e. In my case e = 0.35. $$\sum_{i=13}^{i=25} {{25}\choose{i}} e^{i}(1-e)^{25-i} $$

I could only simplify it to following expression but not of much use. $$1 - \sum_{i=0}^{i=12} {{25}\choose{i}} e^{i}(1-e)^{25-i} $$

The value comes out as ~0.06 using computer code for e = 0.35. Is there any way to evaluate the expression without resorting to programming. A close guess would be also appropriate. This is the link to a similar question, though I couldn't adapt it to solve mine.

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  • $\begingroup$ You could approximate the Binomial distribution by the Normal distribution. Let $X \sim Binomial(25,e)$. $\frac{X-25e}{\sqrt{25e(1-e)}} \approx N(0,1)$. $\endgroup$ – madprob Feb 21 '17 at 17:16
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Your problem is known to be intractible, that's the kicker with Binomial random variables. As you've already shown you can always sum over the complement of what you're interested in, so if that's a much smaller set it's easier, but that's the only trick we know.

As an approximation, we know a Binomial is just a sum of iid Bernouilli r.v.'s, so for large $n$ ($n=25$ may be large enough, it's a judgement call) we can apply the Central Limit Theorem to replace the Binomial with a $\text{Normal}(np,np(1-p))$. It is then considered good manners to find the probability that this Normal r.v. is greater than 11.5 rather than 12, to reconcile the discrepancy between the discrete Binomial and continuous Normal.

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